A346065 - OEIS

%I #24 Jul 29 2021 12:38:06

%S 1,2,8,59,565,6046,68878,818276,10021910,125629220,1603943486,

%T 20783993414,272641113110,3613484662965,48313969712685,

%U 650888627139801,8826840286257595,120398870546499685,1650711840886884265,22735860619151166130,314441081323870331656

%N a(n) = Sum_{k=0..n} binomial(6*k,k) / (5*k + 1).

%C Partial sums of A002295.

%C In general, for m > 1, Sum_{k=0..n} binomial(m*k,k) / ((m-1)*k + 1) ~ m^(m*(n+1) + 1/2) / (sqrt(2*Pi) * (m^m - (m-1)^(m-1)) * n^(3/2) * (m-1)^((m-1)*n + 3/2)). - _Vaclav Kotesovec_, Jul 28 2021

%H Seiichi Manyama, <a href="/A346065/b346065.txt">Table of n, a(n) for n = 0..856</a>

%F G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x)^5 * A(x)^6.

%F a(n) ~ 2^(6*n + 6) * 3^(6*n + 13/2) / (43531 * sqrt(Pi) * n^(3/2) * 5^(5*n + 3/2)). - _Vaclav Kotesovec_, Jul 28 2021

%t Table[Sum[Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 20}]

%t nmax = 20; A[_] = 0; Do[A[x_] = 1/(1 - x) + x (1 - x)^5 A[x]^6 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

%o (PARI) a(n) = sum(k=0, n, binomial(6*k, k)/(5*k+1)); \\ _Michel Marcus_, Jul 28 2021

%Y Cf. A002295, A014137, A104859, A345367, A345368, A346648, A346671, A346672.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Jul 28 2021